A map of sets \(\pi\colon A\to B\) is a surjection if for every \(b\in B\) there exists \(a\in A\) with \(\pi(a)=b\). A right inverse of \(\pi\) is a map \(f\colon B\to A\) with \(\pi\circ f=\id_B\).
Assuming the axiom of choice, a map of sets \(\pi\colon A\to B\) is a surjection if and only if it has a right inverse.
In a category, a morphism with a right inverse is a split epimorphism. A continuous surjection need not have a continuous right inverse: \(t\mapsto e^{2\pi it}\) from \(\RR\) onto \(S^1\) has none, since a continuous \(f\colon S^1\to\RR\) with \(e^{2\pi if(z)}=z\) would make \(\id_{S^1}\) factor through the contractible space \(\RR\) and so be nullhomotopic.